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TrigTable
2005 September 26
A Guide to Evaluating Trigonometric Functions for Common Angle Values
A student is often faced with the task of evaluating one of the six common trigonometric functions for some common multiple of pi,usually integer multiples of
π^6
π^4
π^2
, or
. A student who has completed precalculus with trigonometry should be able to
produce an exact evaluation of the trigonometric function for these angles
without resorting to the use of a calculator
. This document
provides a simple methodology for producing such evaluations.The table on the next page represents our goal – the student should be able to readily reconstruct this somewhat intimidating table. Ablank table is provided on the following page so the student has a template for practice.The remainder of this document walks the student through the elementary steps that allow him to reconstruct the values of the sixtrigonometric for the common angle values of the first four quadrants.
TrigTable
2005 September 26
VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES
π^6
π^4
sin
cos
tan
Udf
Udf
cot
Udf
Udf
Udf
sec
Udf
Udf
csc
Udf
Udf
Udf
Udf means
Undefined
TrigTable
2005 September 26
How to Learn the Table of Trigonometric Values The table has 6 rows and 18 columns (the
column is a repeat of the
0 column) for a total of 144 table values. It may seem
to be a superhuman effort to memorize such a table. Indeed, it is much better to learn the patterns present in the table and use thesepatterns to reconstruct individual entries. The remainder of this document will help you to learn the table patterns.We first notice that the tangent, cotangent, secant, and cosecant functions are derived from the sine and cosine functions. Therefore, ifwe learn the first two table rows, we will be able to reconstruct the remaining four rows. We have cut our work by nearly 1/3! (I usethe word “nearly” because there is some arithmetic involved in calculating the remaining values.)We next notice that the cosine function takes the same values as the sine function, but the values are “shifted” with respect to the angle θ
. If we learn that pattern, it suffices to learn just the first row of the table. Next, we observe that the sine function repeats its values from quadrant to quadrant, with the occasional change of sign. Since all fourquadrants are represented, it suffices to remember the values of the sine function for only the first quadrant.Therefore, if we can remember the sine function for 5 values of
along with some rules for populating the remainder of the table, we
have all 144 table values!There is one more bit of work I neglected to mention – it is necessary to remember the
common values
of the angle parameter
However, patterns once again come to our rescue; it is necessary to learn only approximately 10 different numbers. Radian Values of Common First Quadrant Angles The first quadrant angles of interest have values of 0,
π^6
π^4
π^3
, and
π^2
; call these the
common first quadrant angles
. These radian
measures correspond to degree values of 0, 30, 45, 60, and 90 degrees, respectively. While it is permissible to interpret radianmeasures in terms of the corresponding degree values, the student should quickly learn to think in terms of radian measure.Note that each first quadrant common angle fraction has
as a numerator. The denominators of the sequence of fractions are
decreasing – exactly what is required for the values of the fractions to form an increasing sequence. The denominators are simpleintegers and must be learned.Note also that the first quadrant common angle values are symmetric about
π^4
in that the values can be paired in such a way so that the
sum of each pair is
π^2
. That is, 0
π^2
π^2
and
π^6
π^2
. What about
π^4
? It can be paired with itself:
π^4
π^4
π^2
TrigTable
2005 September 26
Values of the Sine in the First Quadrant The following table shows a simple pattern for remembering the values of the sine function for the angle values described in the priorsection.
π^6
π^4
π^3
π^2
sin
sin
Note that the values in the second row for the sine function have the same value as the corresponding value in the first row. Therefore,if one can begin at 0 and count whole numbers to the value 4, one has everything required to reproduce values of the sine function forthe common first quadrant angle values.
TrigTable
2005 September 26
Values of the Tangent for the Common Angles in the First Quadrant The tangent function is defined as the ratio of the sine and cosine functions. This makes extending the table to include the values forthe tangent function in the first quadrant relatively simple:
π^6
π^4
π^3
π^2
sin
cos
tan
Udf
It may be instructive to review the arithmetic required for rationalization of the denominator. The arithmetic for tan
π^6
is developed:
tan
π^6
Note that the three tangent values for
π^6
π^4
, and
π^3
form a geometric sequence with
3 as the common ratio; that is,
tan
3
tan
π
π
and
tan
tan
π
π
. Note also that
is not in the domain of the tangent function as division by 0 is not
permitted (Udf means undefined.)
TrigTable
2005 September 26
Extending the Table to All Four Quadrants The first step in extending the table to quadrants II, III, and IV is determining the values of the common angles for those quadrants. Asit happens, these values can be easily derived from the corresponding first quadrant common angle values. The portion of the tablethat lists the angle values appears below:
π^6
π^4
π^2
π
Quadrant Boundaries The quadrant boundaries appear at 0,
π^2
, and
π
. Note that the sequence of denominators, 6, 4, and 3, repeats within each
quadrant, but the pattern reverses – descending to ascending to descending… - at each quadrant boundary. Thus, the student should beable to partially reconstruct the first line of the table as follows:
π^6
π^4
π^2
π
Second Quadrant We previously noted the coefficient of pi in the numerator was 1 for the first quadrant common angles. There are similar patterns foreach of the three remaining quadrants.The coefficient of pi in the numerator of the second quadrant common angles is always one less than the value of the denominator.That is, for
we have 2 = 3 – 1. For
we have 3 = 4 – 1, etc. It is a simple matter to complete the second quadrant common
angle values.
π^6
π^4
π^2
π
TrigTable
2005 September 26
Values of the Sine, Cosine, and Tangent Functions for All Four Quadrants The values of the sine, cosine and tangent functions are readily extended to the remaining three quadrants by keeping track of the signof each function in the respective quadrants. There is a simple mnemonic device for remembering which of the three functions ispositive in each of the four quadrants: ASTC (or All Students Take Calculus). Each of the four letters represents one quadrant, A for I,S for II, T for III, and C for IV. The A mean
All
- all three functions are positive in the first quadrant. S represents the sine function –
only the sine function is positive in Quadrant II. T represents the tangent function – only the tangent function is positive in QuadrantIII. Finally, C represents the cosine function – only the cosine function is positive in Quadrant IV.
π^6
π^4
π^2
π
sin
cos
tan
Udf
Udf
What happens if one should not remember a value of one of these functions for a common angle value beyond the first quadrant? Wemay use practice of determination of sign (ASTC) and reference angle to mentally calculate sine, cosine and tangent values forquadrants II, III, and IV as the following examples illustrate.Example 1: Calculate
sin
π
. Since
π
π
π
, we know that
lies in the third quadrant. Using the ASTC mnemonic, we know
sin
π
. The reference angle for
is
π
- the first quadrant common angle with the same denominator. Therefore,
sin
sin
TrigTable
2005 September 26
Example 2: Calculate
tan
. First, verify that
lies in the fourth quadrant. Therefore,
tan
. The reference angle is
π^3
, so
tan
tan
Example 3: Calculate
cos
. Verify that
lies in Quadrant IV so that
cos
. Therefore,
cos
cos
Completing the Table Quite frankly, few people have rapid recall of the values in the bottom half of the table. That is, most mathematicians are much morefamiliar with the values of the sine, cosine, and tangent functions than they are with the cotangent, secant, and cosecant functions.However, every mathematician can readily compute the values given their knowledge of the top half of the table.This is because each value in the lower half of the table is a reciprocal of a corresponding value in the upper half of the table. The onlychallenge is to occasionally rationalize a denominator. Computation is reduced to using function definition, determination of sign byidentification of quadrant, identification of reference angle, and computation of the sine, cosine, or tangent of the reference anglevalues. This is only one more step than what was required for the upper half of the table. The following example illustrates theseprinciples.Example 1: Calculate
sec
. Note that
lies in the second quadrant and that the secant function is the reciprocal of the cosine
function. Therefore, the secant function and the cosine function have the same sign in Quadrant II. By ASTC, the cosine function isnegative in the second quadrant. Therefore,
sec
. Since
is the reference angle, we have
sec
sec
. If the
students fails to remember that
sec
, he or she will remember that
cos
sec
